Mathematisches Institut der Universit¨at M¨unchen
Prof. Otto Forster
May 11, 2016
Riemann Surfaces
Problem sheet #4
Problem 13
Prove that every holomorphic map f : P
1 → C/Λ of the Riemann sphere to a torus is constant.
Hint. Use that P
1 is simply connected.
Problem 14
Let X, Y, Z be locally compact Hausdorff spaces,f :X →Y, g :Y →Z continuous maps and h:=g◦f :X →Z the composite map.
a) Which of the following implications are true, which are false?
i) f and g proper =⇒ h proper, ii) f and h proper =⇒ g proper, iii) g and h proper =⇒ f proper.
Give proofs or counter examples.
b) How does the situation change, if f and g are additionally supposed to be surjective?
Problem 15
a) Show that every root z ∈C of the polynomial F(T) :=Tn+a1Tn−1+. . .+an−1T +an∈C[T] satisfies the estimate |z|62 max{|ak|1/k : 16k 6n}.
b) Let Φ :Cn→Cn be the mapping defined by Φ(z1, . . . , zn) := (sk(z1, . . . , zn))16k6n,
where sk are the elementary symmetric polynomials.
Prove that Φ is a proper, surjective map.
p.t.o.
Problem 16
Let p:Y →X be ann-sheeted branched holomorphic covering map of compact Riemann surfaces X, Y. The trace map
Tr = TrY /X :M(Y)→ M(X)
is defined as follows: For a meromorphic function f ∈ M(Y), let Tr(f) be the first elemen- tary symmetric function of f with respect top, as defined in the course.
Show that all elementary symmetric functions of f with respect to p can be expressed polynomially in terms of Tr(f),Tr(f2), . . . ,Tr(fn).
Give explicit formulas in the cases n= 2 andn = 3.
